无逐点傅里叶衰减的凸曲线的闵可夫斯基和

中文摘要 AI 辅助

设Γ是ℝ²中的紧凸图，定义T(Γ) = inf{ t : 对每个紧集E⊂ℝ²，若Hausdorff维数dim_H(E)>t，则|E+Γ|>0 }。对于在正长度区间上的图，最小可能值为T(Γ)=1。本文研究当弧长的逐点傅里叶衰减不可用时，该最优结论是否仍成立，答案是肯定的，甚至对严格凸曲线也成立。本文采用傅里叶变换约定：ν̂(ξ)=∫e^{-2πi x·ξ}dν(x)。本文构造了一个严格凸的Lipschitz图Γ，满足T(Γ)=1，且对每个非平凡子弧Γ₀和每个α>0，有limsup_{|ξ|→∞}|ξ|^α |Ĥ¹|_{Γ₀}(ξ)|=∞。本文还给出一个凸例子，其每个非平凡子弧上的弧长甚至不是Rajchman测度。这些例子背后的几何机制是正曲率迹：若Γ包含正长度的C²曲线子集，且其曲率远离零，则当dim_H(E)>1时，|E+Γ|>0；对于非退化图，这给出T(Γ)=1；对于凸图，这特别意味着当曲率测度具有非零绝对连续部分时，T(Γ)=1。正测度的证明在物理空间中进行，使用平移管的交集和凸函数的基本事实，相同的重叠估计在正曲率迹假设下给出相关曲线投影邻域平均长度的Mattila型下界。本文还证明了具有正长度可求长部分的集合的一维端点结果，并提出了主要剩余问题：是否每个严格凸Lipschitz图都具有最优阈值T(Γ)=1。

英文摘要

Let $Γ\subset\mathbb R^2$ be a compact convex graph and define \[ T(Γ) = \inf \left\{ t: \dim_{\mathrm H}(E)>t \Longrightarrow |E+Γ|>0 \text{ for every compact }E\subset\mathbb R^2 \right\}. \] For a graph over an interval of positive length the smallest possible value is $T(Γ)=1$. We ask whether this optimal conclusion can hold when pointwise Fourier decay of arclength is unavailable. The answer is yes, even for strictly convex curves. We use the Fourier transform convention $\widehatν(ξ)=\int e^{-2πi x\cdotξ}\,dν(x)$. We construct a strictly convex Lipschitz graph $Γ$ with $T(Γ)=1$ such that, for every nontrivial subarc $Γ_0$ and every $α>0$, \[ \limsup_{|ξ|\to\infty} |ξ|^α\left| \widehat{H^1|_{Γ_0}}(ξ) \right|= \infty. \] We also give a convex example for which arclength on every nontrivial subarc fails even to be a Rajchman measure. The geometric mechanism behind these examples is a positive curved trace: if $Γ$ contains a positive-length subset of a $C^2$ curve whose curvature is bounded away from zero, then $|E+Γ|>0$ whenever $\dim_{\mathrm H}(E)>1$. For a nondegenerate graph this gives $T(Γ)=1$. For convex graphs it implies, in particular, that $T(Γ)=1$ whenever the curvature measure has a nonzero absolutely continuous part. The positive-measure proofs are in physical space and use translated-tube intersections and elementary facts about convex functions. The same overlap estimates give Mattila-type lower bounds for the average lengths of the associated curve projections of neighborhoods under the positive curved-trace hypothesis. We also prove a dimension-one endpoint result for sets with a positive-length rectifiable part and formulate the main remaining question: whether every strictly convex Lipschitz graph has the optimal threshold $T(Γ)=1$.

发表机构

• University of Rochester（罗切斯特大学）
• The Ohio State University（俄亥俄州立大学）
• Virginia Tech（弗吉尼亚理工大学）
• University of British Columbia（不列颠哥伦比亚大学）

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